Now, we are in the cohomogeneity zero case of the reduction theory introduced in [1]}. Then, according to [2]}, the map \(\Phi _{\eta ,\nu }\)  is biharmonic if and only if \((\eta , \nu )\)  is a critical point of \(\hat{E}_2\) , that is
\(\frac{\partial \hat{E}_2}{\partial \eta }(\eta ,\nu )=0 \quad {\rm and}\quad \frac{\partial \hat{E}_2}{\partial \nu }(\eta ,\nu )=0 \,.\) 
